# Center for Waves in Stratified Media

## Introduction

Scattering is a natural phenomenon. Looking at the sky, we see scattered sunlight. An important fact in wave scattering is that a wave is not scattered when it traverses a homogeneous medium. Only inhomogeneities cause scattering. Therefore, scattering is also a tool to explore inhomogeneities in a medium.

Scattering theory is concerned with the effect an inhomogeneity has on an incident wave or particle. In general, there are two kinds of problems in scattering: direct scattering and inverse scattering. We consider the total wave field as the sum of an incident wave and a scattered wave. The direct scattering problem is to determine the scattered wave from the knowledge of the incident wave and the known inhomogeneity. The inverse scattering problem, which is more interesting and more challenging, is to determine the unknown inhomogeneity from the knowledge of the incident wave and the measured scattered wave.

The direct scattering problem has been studied for many years. The problems are formulated as a boundary value problem, i.e., an equation called partial differential equation along with a set of boundary conditions. Solving the problem we can predict how waves propagate in a given medium. However, scientists can obtain solutions analytically only for very simple media. For inverse problems, analytic solutions are almost impossible.

This situation was dramatically changed in the last a few decades due to the development of powerful computers. Some stable numerical methods for solving complicate boundary problems have been developed. Solving corresponding inverse problems is possible. These lead to extremely active study of the inverse scattering problems.

Our research is to investigate systematically the inverse scattering of acoustic and electromagnetic waves by obstacles or inhomogeneity in a stratified host medium. By stratified host medium we mean that the refractive index (a quantity for describing the inhomogeneity of the medium) of the medium is a function of one variable except in a bounded domain.

The basic problems we consider are the inverse scattering of time-harmonic acoustic or electromagnetic waves by a penetrable non-stratified inhomogeneous medium of compact support and by a bounded impenetrable obstacle in a stratified host medium. We also study inverse scattering problems in a fluid-elastic/fluid-poroelastic sediment. The problems can be generally stated as following: given the scattered waves for several incident waves with different incident directions and different modes, determine the shape of the scatterer or the inhomogeneous refractive index function.

The methodology used by Dr. R. Gilbert and Dr. Y. Xu for the reflecting sea bottom was based on finding an operator which produced the far-field from a scattered wave. This method can be generalized to more complicated problems.

The Significance of our research may be seen from different aspects. In the application aspect, our research is motivated by the need to reconstruct the shape of scattering object and determine its refractive index function from scattered data in a stratified host medium. This need can be found in many pressing issues, including the following.

1. Environmental protection.

Temperature is a major factor to affect the refractive index of a medium. Global or large area change of temperature will affect the propagation of acoustic waves in oceans. Measuring the propagation and scattering of acoustic waves may provide an important way to determine the long term trend of temperature change. Our research can provide theoretical analysis and computer simulations for determination of the refractive index from measured acoustic waves.}

2. Exploration of oceans.

The oceans cover more than half of the earth's surface. Their depths are filled with an unlimited variety of things we are interested in detecting, measuring, interpreting, analyzing, modeling, exploiting, and preserving. Sound wave is the only energy form that can propagate effectively in water. A better understanding of the inverse scattering of underwater sound waves will greatly improve ocean survey systems.

3. Remote sensing of objects under ice cover.

Electromagnetic waves are used in remote sensing of areas with ice cover. Inverse scattering theory of electromagnetic waves will provide more effective algorithms in computerized detection.